Chapter 3 Answers to Problems

Transcription

1 Chapter 3 Answers to Problems 3.1 (a) a = A 1 + B 2 + E (b) b = 3A 1 + A 2 + 4E (c) c = 2A 1' + E' + A 2" (d) d = 4A 1 + A 2 + 2B 1 + B 2 + 5E (e) e = A 1g + A 2g + B 2g + E 1g + 2E 2g + A 2u + B 1u + B 2u + 2E 1u + E2u (f) f = A 1 + E + T 1 + 3T2 (g) g = A 1g + E g + T 1g + T 2g + 3T 1u + T2u (h) h = 2A 1' + E 1' + E 2' + A 2" 3.2 (a) a = 4A + 3B + 4{E} * (b) b = 4A' + 5{E'} + 3A" + 2{E"} [Note: + = 2 cos 2 /3 = 2(-0.5) = -1] * 2 *2 (c) c = 3A + {E 1} + {E 2} [Note: + = 2 cos 2 /5 = , and + = 2 cos 4 /5 = ] 3.3 (a) C 4v C4 C4v C z, x +y, z A A z, x +y, z, R z Rz A x -y B1 B x -y, xy xy B 2 (x,y), (R x,r), y (xz, yz) E {E} (x,y), (R x,r), y (xz, yz) 17

2 (b) D 3h D3 D3h D x +y, z A 1' A1 x +y, z, ( ) Rz A 2' 2 2 (x,y), (x -y, xy) E' A2 z, Rz A" z A 2" E (x,y), (x -y, xy), (R,R), x y (xz, yz) (R,R), (xz, yz) E" x y (c) D C 5d 5v D 5d C 5v x +y, z A1g A1 z, x +y, z R z A 2g (xz, yz), (R x,r) y E1g A2 R, z ( ) 2 2 (x -y,xy) E 2g A1u E 1 (xz, yz), (R x,r) y (x,y) z A 2u 2 2 (x,y) E1u E 2 (x -y,xy), ( ) E 2u 18

3 3.4 D 4h D 2d, where 2C 2' of D 4h 2C 2' of D 2d, and 2 d of D 4h 2 d of D 2d. (Columns for operations of D 4h that are not shared by this definition of D 2d have been omitted in the table below.) D4h E 2S4 C2 2C 2' 2 d E 2S4 C2 2C 2' 2 d D2d A1g A1 A2g A2 B1g B1 B2g B2 Eg E A1u B1 A2u B2 B1u A1 B2u A2 Eu E 19

4 D 4h D 2d, where 2C 2" of D 4h 2C 2' of D 2d, and 2 v of D 4h 2 d of D 2d. (Columns for operations of D 4h that are not shared by this definition of D 2d have been omitted in the table below.) D4h E 2S4 C2 2C 2" 2 v E 2S4 C2 2C 2' 2 d D2d A1g A1 A2g A2 B1g B2 B2g B1 Eg E A1u B1 A2u B2 B1u A2 B2u A1 Eu E (a) In C 2v, a = 5A 1 + 5B 1 + 5B 2, from which it follows in C v, a = (b) In D, = A + B + B + 2B + 2B + 2B, from which it follows in D, = h b g 2g 3g 1u 2u 3u h b g + g u u 3.6 (a) In D 4d, B 2 B 2 = A 1 (totally symmetric representation) [Rule 5, p. 83] (b) In T d, for T 2 T 2, Rules 4 and 5 apply. (c) In D 6d, A 1 E 5 = E 5 by Rules 3 and 6. Td E 8C3 3C2 6S4 6 d T T = A 1 + E + T 1 + T2 20

5 (d) In D 2d, for B 1 B 2, Rules 1 and 6 apply. D2d E 2S4 C2 2C 2' 2C 2" B B = A (e) In C 4h, for B g A u, Rules 1 and 6 apply. Also, note g u = u. 3 3 C2h E C4 C2 C4 i S4 h S4 Bg Au = Bu (f) In D 3h, for A 1" A 2", Rules 1 and 6 apply. Also note " " = '. D3h E 2C3 3C2 h 2S3 3 v A 1" A 2" = A 2' (g) In C 4h, for A u E u, Rules 2 and 6 apply. Also note u u = g. 3 3 C4h E C4 C2 C4 i S4 h S4 Au {E u} = {E g}

6 (h) In D 3d, for E g E u, Rules 4 and 6 apply. Also note g u = u. D3d E 2C3 3C2 i 2S6 3 d E g Eu = A 1u + A 2u + Eu 3.7 (a) (i) D 3h, C 2v, C s (ii) I II: lost C 3, 2C 2, S 3, 2 of 3 v (the retained v becomes v', and the becomes in C ); II III: lost C, (or ') (iii) descent (iv) yes (v) yes h v 2v 2 v v (b) (i) T d, C 3v, C 2v (ii) I II: lost 3 of 4C 3, 3C 2, 3S 4, 3 d (the other 3 d are retained in C 3v); II III: lost C 3 and 2 of 3 v (one is retained as either v or v' in C 2v), and gained C 2 and another (iii) descent (iv) no (v) yes v (c) (i) D 3d, C s, C 3v (ii) I II: lost C 3, 3C 2, i, S 6, and 2 of 3 d (one d is retained as h of C s); II III: regained C and 2 (one other is from C ) (iii) ascent (iv) yes (v) yes 3 v v h s (d) (i) D 3d, C s, C 2v (ii) I II: lost C 3, 3C 2, i, S 6, and 2 of 3 d (one d is retained as h of C s); II III: gained C and another (other is of C ) (iii) ascent (iv) yes (v) no 2 v v h s 3d 3 3h 6 d h 3 v (e) (i) D, D, D (ii) I II: lost i, S, and 3 ; II III: gained, S, and 3 (iii) ascent (iv) yes (v) no 22

7 3.8 (a) D 3h C 2v C s. Use correlation tables in Appendix B to construct the following correlations. D3h C2v Cs A' 1 A' 2 A1 A' E' A 2 A" 1 B1 A" 2 B2 A" E" The E' and E" degeneracies are lifted in the descent D C. (b) T d C 3v. Use correlation tables in Appendix B to construct the following correlations. Note that there is no correlation between C 3v and C 2v, because they do not have a groupsubgroup relationship. 3h 2v T d C 3v A1 A1 A 2 E T1 A 2 E T 2 The T 1 and T 2 triple degeneracies are partially lifted to become a nondegenerate and doubly degenerate species in C. The double degeneracy of E is retained. 3v 23

8 (c) D 3d C s C 3v. The C s C 3v correlations can be obtained from the C 3v correlation table. The D 3d C s correlations can be deduced by matching vector transformations in the two groups. However, the axes orientations change on the descent D 3d C s. the z axis of C s is either the x or y axis of D 3d. If we assume that it is x of D 3d, then the descent causes the following axis transformations: z x, x z, y y. Thus A 2g (R z) correlates with A" (R x), etc. A 1u connects to A", because A" must connect to A 2 in C 3v to establish the overall correlation A 1u A 2 of D 3d C 3v, as listed in the correlation table. The check of the D 3d C s correlations is that they all provide a continuous path that makes the listed D 3d C3v correlations. D3d Cs C3v A 1g A 1 A 2g A' E g A 2 A 1u A" A 2u E E u The E g and E u degeneracies are lost in the D 3d C s descent, but A' and A" become degenerate as E in the ascent C C. s 3v (d) D 3d C s C 2v. As in (c), there is an axis shift with D 3d C s. The correlations shown for D 3d C s are the same as shown in (c) (see above for explanation). Assuming z x, x z, y y in the D 3d C s descent, the h plane of C s is in the yz plane of the D 3d coordinate system. The conventional orientation of the coordinate system in C 2v is the same as in D 3d. Therefore, on the ascent C s C 2v, h yz. The correlations shown below are based on h, taken from the C correlation table. yz 2v 24

9 D3d Cs C2v A1g A1 A 2g E g A' A 2 A 1u 1 A" A 2u B 2 E u B The E g and E u degeneracies are lost in the D 3d C s descent. Note that there is no direct correlation between D 3d and C 2v, because they do not have a group-subgroup relationship. Nonetheless, they can be linked through their shared subgroup, C. (e) D 3d D 3 D 3h. Use correlation tables in Appendix B to construct the following correlations. D 3d D3 D3h A A A ' 1g 1 A 1 2g 2 A ' s Eg E' A 2 A A " 1u 1 A2u A 2" E E E" u All degeneracies are retained, and no new degeneracies are established through the transformations. Note that there is no direct correlation between D 3d and D 3h, because they do not have a group-subgroup relationship. Nonetheless, they can be linked through their shared subgroup, D 3 25

10 3.9 D 4 C4 D4 E 2C4 C2 2C 2' 2C 2" 3 E C 4, C4 C C4 A A A A B B B B E {E} D 4 C 2 (C 2 = C 2 4 ) D4 E 2C4 C2 2C 2' 2C 2" C2 A A A A B A B A E B D 4 C 2 (C 2 = C 2') D4 E 2C4 C2 2C 2' 2C 2" C2 A A A B B A B B E A + B 26

11 D 4 C 2 (C 2 = C 2") D4 E 2C4 C2 2C 2' 2C 2" C2 A A A B B B B A E A + B 3.10 There are several ways of doing this, including the following procedure. For the initial I T correlation, use matching of vector transformations in the two groups and also the method of examining characters of operations of I retained in T. Once the I T correlation has been established, use the tabulated correlations for T to make correlations to groups that are subgroups of both I and T. For example, for I C 3, use the I T correlations and then the tabulated T C 3 correlations to make the links. Likewise, for I D 2, use the I T correlations and then the tabulated T D 2 correlations to make the links. For I C 2, use the previous results for D 2 and make use of the tabulated D 2 C 2 correlations to make the links. For both I D 5 and I D 3, use matching of vector transformations in I and the subgroup and also the method of examining characters of operations of I retained in the subgroup. For I C 5, use the I D 5 results and the tabulated D 5 C 5 correlations to make the links. 27